Final answer:
To calculate the gauge pressure in a tapered pipe system for water to emerge at a given velocity, apply the Bernoulli's equation considering the conservation of energy in a fluid and the continuity equation to account for the changes in velocity and area.
Step-by-step explanation:
The question pertains to the gauge pressure required in a tapered pipe system for water to emerge from the narrower end at a certain velocity. To solve this problem, one needs to apply the principles of fluid mechanics, specifically the Bernoulli's equation, which relates the velocities, pressures, and heights at two points within a flowing fluid. According to Bernoulli's principle, the equation P1 + 0.5∙ρ∙v1^2 + ρ∙g∙h1 = P2 + 0.5∙ρ∙v2^2 + ρ∙g∙h2 applies, where ρ is the density of water, P is the pressure, v is the velocity, g is the acceleration due to gravity, and h is the height above a reference level. Since the diameters of the ends are given as being in a 2:1 ratio, and area scales with the square of the diameter, the area ratio is 4:1. We can use the continuity equation A1∙v1 = A2∙v2 to relate the velocities at the two ends. Finally, we can solve for the gauge pressure (P1 - P2), taking into account the height difference.